Fixed points for scalar theories in 4 − ε, 6 − ε and 3 − ε dimensions are discussed. It is shown how a large range of known fixed points for the four dimensional case can be obtained by using a general framework with two couplings. The original maximal symmetry, O(N ), is broken to various subgroups, both discrete and continuous. A similar discussion is applied to the six dimensional case. Perturbative applications of the a-theorem are used to help classify potential fixed points. At lowest order in the ε-expansion it is shown that at fixed points there is a lower bound for a which is saturated at bifurcation points. arXiv:1707.06165v7 [hep-th]

There is a widely held belief that conformal field theories (CFTs) require zero beta functions. Nevertheless, the work of Jack and Osborn implies that the beta functions are not actually the quantites that decide conformality, but until recently no such behavior had been exhibited. Our recent work has led to the discovery of CFTs with nonzero beta functions, more precisely CFTs that live on recurrent trajectories, e.g., limit cycles, of the beta-function vector field. To demonstrate this we study the S function of Jack and Osborn. We use Weyl consistency conditions to show that it vanishes at fixed points and agrees with the generator Q of limit cycles on them. Moreover, we compute S to third order in perturbation theory, and explicitly verify that it agrees with our previous determinations of Q. A byproduct of our analysis is that, in perturbation theory, unitarity and scale invariance imply conformal invariance in four-dimensional quantum field theories. Finally, we study some properties of these new, "cyclic" CFTs, and point out that the a-theorem still governs the asymptotic behavior of renormalization-group flows.

It's well known that in conformal theories the two-and three-point functions of a subset of the local operators-the conformal primaries-suffice, via the operator product expansion (OPE), to determine all local correlation functions of operators. It's less well known that, in superconformal theories, the OPE of superdescendants is generally undetermined from those of the superprimaries, and there is no universal notion of superconformal blocks. We recall these and related aspects of 4d (S)CFTs, and then we focus on the super operator product expansion (sOPE) of conserved currents in 4d N = 1 SCFTs. The current-current OPE J(x)J(0) has applications to general gauge mediation. We show how the superconformal symmetry, when combined with current conservation, determines the OPE coefficients of superconformal descendants in terms of those of the superconformal primaries. We show that only integer-spin real superconformal-primary operators of vanishing R-charge, and their descendants, appear in the sOPE. We also discuss superconformal blocks for fourpoint functions of the conserved currents.July 2011 arXiv:1107.1721v2 [hep-th] 30 Apr 2014(1.4) with notation reviewed in section 3.1. For now we will just say that X −X = 4iΘΘ, withto superconformal primary components, but do contribute for superdescendants. Explicitly, in (1.3), the f abc term is a descendant coefficient that is unrelated to the kd abc primary coefficient. In (1.4) the Θ dependence is at least determined by G symmetry. For general operators, the Θ dependence is ambiguous, not fully determined by the symmetries.We will here study the general constraints of superconformal symmetry on the twoand three-point functions relevant for the J(x)J(0) sOPE, and how the sOPE coefficients are obtained from these correlators. We will do this both using the superspace results of Osborn [11] for the relevant two-and three-point functions, and also directly from the superconformal algebra. As we'll discuss, the fact that the currents are conserved here allows the superspace Θ dependence to be completely fixed. Thus, the coefficients of the superconformal primaries in the J(x)J(0) OPE suffice to fully determine all OPE coefficients of all descendants. We will discuss the contributions on the RHS of the J(x)J(0)OPE from integer-spin real U (1) R -charge-zero superconformal primaries, O µ 1 ...µ , and their superdescendants. 3The paper is organized as follows: section 2 briefly reviews the aspects of the OPE in 4dCFTs that we will use in the following discussion. Section 3 discusses superconformal theories, and the constraints of superconformal symmetry on two-and three-point functions and the OPE. The superspace formalism of [11], and the recent results about chiral-chiral and 3 Note added (April 2014 revision): as was later found in [15], there are additional contributing Lorentz representations. This revised version will also correct a couple of errors in our original version's coefficients, as pointed out to us by the authors of [15] and [16]; see these papers for further ...

Fixed points of scalar field theories with quartic interactions in d = 4 − ε dimensions are considered in full generality. For such theories it is known that there exists a scalar function A of the couplings through which the leading-order beta-function can be expressed as a gradient. It is here proved that the fixed-point value of A is bounded from below by a simple expression linear in the dimension of the vector order parameter, N . Saturation of the bound requires a marginal deformation, and is shown to arise when fixed points with the same global symmetry coincide in coupling space.Several general results about scalar CFTs are discussed, and a review of known fixed points is given.Dedicated to the memory of Louis Michel , the first IHES professor of physics and a pioneer of group theory applications to RG flows.